Prove that the Product of Three Consecutive Positive Integers is Divisible by 6

By K Balaji|Updated : November 13th, 2022

It is proved that the product of three consecutive positive integers n (n + 1) (n + 2) is divisible by 6

If three consecutive numbers n, n + 1, and n + 2 are used, then.

When a number is divided by 3, the result is always one of the following: 0, 1, or 2.

Let n equal 3p, 3p + 1, or 3p + 2, with p being an integer.

If n = 3p, then n is divisible by 3.

If n = 3p + 1, then n + 2 = 3p + 1 + 2 = 3p + 3 = 3(p + 1) is divisible by 3.

If n = 3p + 2, then n + 1 = 3p + 2 + 1 = 3p + 3 = 3(p + 1) is divisible by 3.

So that n, n + 1 and n + 2 is always divisible by 3.

⇒ n (n + 1) (n + 2) is divisible by 3.

Similar to this, whenever a number is divided by 2, the result is either 0 or 1.

∴ n = 2q or 2q + 1, where q is some integer.

If n = 2q, then n and n + 2 = 2q + 2 = 2(q + 1) are divisible by 2.

If n = 2q + 1, then n + 1 = 2q + 1 + 1 = 2q + 2 = 2 (q + 1) is divisible by 2.

So that n, n + 1 and n + 2 is always divisible by 2.

⇒ n (n + 1) (n + 2) is divisible by 2.

But n (n + 1) (n + 2) is divisible by 2 and 3.

∴ n (n + 1) (n + 2) is divisible by 6.

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